BAGHDAD – Amid hedgerows of computer screens in the joint operations center that runs the war against the Islamic State, Marine Brig. Gen. Bill Mullen explains the complex assault that drove the extremist fighters last week from the strategic town of Rutbah at the western edge of Anbar province.

The battle showed how the campaign against the Islamic State, which has had a slow takeoff over the last 18 months, is supposed to work: In early May, a U.S. drone attack on a nearby highway killed Shaker Wahib, the terrorists’ military “emir” in Anbar, shaking morale. The day before the battle, the U.S. dropped two huge bombs on the minefields and berms surrounding the town.

Then came the attack from a combined force of Iraqi army troops and hundreds of recently recruited tribal fighters who had been trained by U.S. Special Forces. When they moved in, only 30 Islamic State fighters stayed to fight, says Mullen. The rest had fled.

Mullen has been running the operations room here for nearly a year. When he arrived, he says, a “serious gloom” enveloped the Iraqi military because of its humiliating defeats in Mosul and Ramadi. But with the recapture of Ramadi late last year, says Mullen, some of that lost confidence has returned.

Iraq is still a nation in disarray, with bitter ethnic hatreds and a central government that has nearly collapsed. The military campaign rests on political quicksand. The fragility was shown by Friday’s invasion of the Green Zone by Iraqi demonstrators who are enraged by the corruption of their government.

Gen. Joseph Votel, the recently appointed Centcom commander who oversees the U.S. military in the Middle East, sees a mixed picture in the Iraqi military. “They are getting better … (but) there’s still a lot left to do,” he told me and several other journalists who traveled here with him.

Lt. Gen. Sean MacFarland, a lanky, blunt-spoken officer who took command of U.S. forces in Iraqi and Syria late last year, explains: “We’ve had to pick (the Iraqi army) off the ground and dust them off. … They are certainly better than the enemy. That’s the standard.”

Votel and MacFarland are trying to accelerate a campaign that had seemed, at times over the last 18 months, to be foundering. They’re more open to the media, as illustrated by our trip here, and they’re working harder to coalesce the elements of the U.S.-led coalition. Their goal is to stress the Islamic State on many fronts at once — preparing assaults on Mosul, Fallujah and other strongholds. The multi-pronged strategy, says Votel, reflects “the virtues of simultaneity.”

The military power that the U.S. can mobilize is daunting: We visited a warehouse packed with 37,000 sets of body armor and 32,000 M-16 rifles that will be airlifted to the Iraqis. We saw command centers that fuse intelligence from scores of drones — and direct fire from missiles, jets, artillery and Special Operations commandos.

We visited a camp in Taji where trainers from coalition nations are attempting to rebuild “an army that lost face” after its collapse in Mosul in June 2014, explains Lt. Col. Jim Hammett. He’s an Australian special forces officer who helps command a 480-person team from Australia, New Zealand and Britain. Amid the endless frustrations of Iraq, the training team has a one-word motto: “perseverance.”

The most striking change is the mobilization of Sunni tribes in Anbar since the Islamic State began to lose its grip in Ramadi last October. The government-paid Sunni militia in Anbar has grown this year from 6,000 to 9,500, and it’s now supplemented by an additional vetted force of 6,000 “tribal shield” fighters, who aren’t paid a salary but get weapons and death benefits. Another 9,000 Anbar tribesmen have volunteered on an unofficial basis.

If more Sunni sheiks are working with the U.S. (and an Iraqi government they despise), it’s for a cynical reason: They think the American side is winning. U.S. commanders name seven Sunni tribes that are now contributing fighters against the Islamic State. What’s intriguing is that some of these tribes are said to be split, with part still backing the Islamic State and others defecting. The U.S. strategy is to treat the so-called caliphate as a weak state — and turn the tables by mounting an insurgency against it from the inside.

America’s military strength remains overwhelming, even after the tests of the past decade, and the emerging campaign almost surely will gradually disable the Islamic State. The problem, as nearly every commander here will acknowledge, is that U.S. military might cannot make a broken Iraq work as a nation.